The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of \(\frac{3}{2}\), even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only \(\frac{4}{3}\). Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (FOCS, 550–559, 2011), and then by Mömke and Svensson (FOCS, 560–569, 2011). In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560–569, 2011) yielding a bound of \(\frac{13}{9}\) on the approximation factor, as well as a bound of \(\frac{19}{12}+\varepsilon\) for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

中文翻译：

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[公式：请参见文字]-图形TSP的近似值。

在近似算法中，旅行商问题是基础问题和深入研究的问题之一。30多年来，对于通用度量标准而言，最好的算法就是Christofides的算法，其近似因子为\（\ frac {3} {2} \），即使问题的所谓Held-Karp LP松弛是推测其完整性差距仅为\（\ frac {4} {3} \）。最近，Oveis Gharan等人首先在图形度量的重要特殊情况方面取得了重大进展。（FOCS，550-559，2011年），然后是Mömke和Svensson（FOC​​S，560-569，2011年）。在本文中，我们对Mömke和Svensson（FOC​​S，560–569，2011）中提出的方法进行了改进分析，得出了\（\ frac {13} {9} \）的界线。近似因子，以及对于任何ε > 0的\（\ frac {19} {12} + \ varepsilon \）的界，对于图形指标中更一般的Traveling Salesman Path问题。